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Coupled Sparse Denoising and Unmixing With Low-Rank Constraint for Hyperspectral Image Jingxiang Yang, Student Member, IEEE, Yong-Qiang Zhao, Member, IEEE, Jonathan Cheung-Wai Chan, and Seong G. Kong, Senior Member, IEEE

Abstract—Hyperspectral image (HSI) denoising is significant for correct interpretation. In this paper, a sparse representation framework that unifies denoising and spectral unmixing in a closed-loop manner is proposed. While conventional approaches treat denoising and unmixing separately, the proposed scheme utilizes spectral information from unmixing as feedback to correct spectral distortion. Both denoising and spectral unmixing act as constraints to the others and are solved iteratively. Noise is suppressed via sparse coding, and fractional abundance in spectral unmixing is estimated using the sparsity prior of endmembers from a spectral library. The abundance of endmembers is used as a spectral regularizer for denoising based on the hypothesis that spectral signatures obtained from a denoising process result are close to those of unmixing. Unmixing restrains spectral distortion and results in better denoising, which reciprocally leads to further improvements in unmixing. The strength of our proposed method is illustrated by simulated and real HSIs with performance competitive to the state-of-the-art denoising and unmixing methods. Index Terms—Coupling, denoising, hyperspectral image (HSI), sparsity, unmixing.

I. I NTRODUCTION

H

YPERSPECTRAL imaging technology provides rich spectral information of a scene, which is superior to those acquired by multispectral imaging methods. Hyperspectral images (HSIs) have generated much research interest in fields such as terrain classification, target detection, environmental monitoring, mineral exploration, and military surveillance [1]–[3], [5], [6]. Due to the dark current from thermal emission and

Manuscript received December 20, 2014; revised June 15, 2015; accepted September 17, 2015. This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 61371152, Grant 61071172, and Grant 61374162; by the NSFC and National Research Foundation of Korea Scientific Cooperation Program under Grant 6151101013; by the New Century Excellent Talents Award Program from the Ministry of Education of China under Grant NCET-12-0464; by the Ministry of Education Scientific Research Foundation for the Returned Overseas; by the Fundamental Research Funds for the Central Universities under Grant 3102015ZY045; and by the China Scholarship Council for joint Ph.D. students under Grant 201506290120. J. Yang and Y.-Q. Zhao are with the Key Laboratory of Information Fusion Technology (Ministry of Education of China), School of Automation, Northwestern Polytechnical University, Xi’an 710072, China (e-mail: [email protected]; [email protected]). J. C.-W. Chan is with the Department of Electronics and Informatics, Vrije Universiteit Brussel, 1050 Brussel, Belgium (e-mail: jcheungw@ etro.vub.ac.be). S. G. Kong is with the Department of Computer Engineering, Sejong University, Seoul 05006, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2015.2489218

the stochastic error of photoelectron counting in the imaging process, HSIs are unavoidably corrupted by noise in the acquisition process [7]. Consequently, it degrades the HSI’s discriminative performance and jeopardizes application results. HSI denoising is a necessary preprocessing step, and denoising performance will influence the subsequent discrimination accuracy. Recently, different HSI denoising methods have been proposed [8]–[19]. The simplest way is to apply the traditional 2-D or 1-D denoising methods to the HSI band by band or pixel by pixel. However, the results of these simple approaches are not satisfying. If denoising is applied independently in the spatial and spectral domains, the natural spectral–spatial correlation will be destroyed, and artifacts may be introduced. Therefore, spatial and spectral information should be considered jointly to make denoising more efficient. Rather than applying traditional total variation (TV) term band by band, a spectral–spatial adaptive TV was used in [9] and [16]. In [11] and [13], dictionary containing spatial–spectral correlations was learned to reconstruct a noise-free HSI. Then, a low-rank constraint was added to exploit global spectral correlation in the process of reconstruction. The low-rank nature is also used to remove mixed noise of HSI in [19]. In [14], an HSI denoising method was proposed via exploiting nonlocal spatial–spectral structural similarity. To enforce spectral smoothness and spatial discontinuity in denoising, the first-order derivative was applied along both the spatial and the spectral dimensions [12]. Similarly, the spectral first-order derivative in the wavelet domain was introduced as a regularizer for HSI denoising [18]. Taking advantage of the statistical differences between signal and noise, noise can be removed efficiently in the wavelet domain. Instead of 2-D wavelet transform, 3-D wavelet transform is used to collaboratively reduce noise in the spectral dimension of HSIs [15], [17]. In [8], principal component analysis (PCA) transform was applied, and then wavelet denoising methods were implemented to nonprincipal components’ spatial 2-D and spectral 1-D dimensions. In the case of low noise levels, spectral derivative was first applied as preprocessing to elevate noise; then, wavelet-based spatial and spectral denoising was implemented [10]. Other than stochastic noise, most pixels are spectrally mixed. Due to limited spatial resolution, intimate mixture, and atmosphere scattering, spectra of more than one endmembers would be contained in one pixel. Unmixing is the process of determining endmember abundance at the subpixel level. The unmixing method can be classified into three categories: geometrical,

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Fig. 1. Flowchart of the traditional HSI processing framework, taking unmixing as an example.

statistical, and sparse regression [4]. Geometric-based unmixing methods often assume the presence of at least one pure pixel per endmember in the scene. This assumption does not hold in most cases in reality. In statistical methods, each endmember and its associated abundance can be estimated using the Bayesian maximum a posteriori estimator, whereas the prior distributions of endmember and abundance are used to regularize the solutions [43], [44]. Alternatively, unmixing can be treated as a sparse regression problem by assuming that spectrum in each pixel can be approximated linearly using few spectral signatures from a given library [31], [33]. Most unmixing methods do not consider inherited noise and this limits their performance. However, a general problem of denoising is spectral distortion, which inevitably imposes negative effects on unmixing. Therefore, the spectral distortion should be restrained in the process of denoising. To this end, we can make use of the spectral information after unmixing. For example, according to the spectral mixture model (linear or nonlinear), the spectrum on each pixel can be recomputed using endmember abundance achieved from unmixing [4]. This spectral information can be used as feedback to suppress spectral distortion. In this paper, we propose a coupled HSI denoising and unmixing method (CHyDU). Denoising and unmixing processes are unified in a sparse representation framework. The premise is that the spectrum on each pixel obtained from the denoised image should be close to the spectrum recreated from unmixing; therefore, the disparity is expected to be small. This small disparity is designed as a spectral regularizer for denoising, which is performed in a sparse representation framework via dictionary learning and sparse coding. With the spectral regularization constraint, denoising will be more efficient, resulting in better endmember abundance, which is estimated via sparse regression by employing the sparse prior of endmember over a given spectral library (e.g., U.S. Geological Survey (USGS) library)1 [33]. The algorithm solves both denoising and unmixing iteratively, benefiting from the intermediate results from each other. The final optimized denoising and unmixing result will be achieved simultaneously. We consider three main contributions of this paper. First, we demonstrate that not only interpretation can be improved by denoising but denoising can also be reciprocally improved by intermediate interpretations in the loop. A coupled denoising– 1 Available

online: http://speclab.cr.usgs.gov/spectral-lib.html.

interpretation algorithm can be formulated in a unified framework, in which denoising and interpretation promotes each other. Second, we introduce spectral unmixing as an example and propose a coupled hyperspectral denoising and unmixing in a sparse representation framework. Third, we show that the disparity between spectral signatures from the denoising result and unmixing result, when exploited as spectral regularization, suppresses the spectral distortion in the process of denoising, resulting in unifying denoising and unmixing and achieving both results simultaneously. The remainder of this paper is organized as follows. In Section II, we will illustrate the idea that denoising and interpretation can benefit from each other if they are unified in one framework. In Section III, by taking unmixing as an example and connecting it with denoising in a unified sparse representation framework, we propose a coupled denoising and unmixing algorithm for HSIs. The solver and computational complexity analysis is presented here. Experiment results and analysis are given in Section IV. We make our conclusions in Section V. II. C OUPLING D ENOISING AND I NTERPRETATION Traditionally, denoising and interpretation are often treated separately. Noise in HSIs is removed prior to interpretation such as spectral unmixing, as shown in Fig. 1. The sequential scheme tends to suffer from spectral distortion caused by denoising. Spectral distortion affects the accuracy of spectral unmixing. Spectral distortion affects the accuracy of spectral unmixing. Artifacts caused by the denoising can propagate to the unmixing step and degrade the performance of interpretation [22]–[24]. The challenges of traditional sequential scheme can be overcome by coupling denoising and unmixing procedures in a unified framework in a closed-loop way, as shown in Fig. 2. In this coupled scheme, both denoising and unmixing can act as constraints for each other, and their performance can be improved simultaneously. Assuming an initial unmixing result is given, then it can be used as feedback to act as a spectral constraint and to restrain the spectral distortion in the coupled scheme. By utilizing a spectral prior constraint, the distortion introduced in denoising can be significantly reduced. At the same time, a more accurate unmixing result can be obtained using the data with less noise. On one hand, the spectral constraint provided by unmixing can give a powerful regularization for denoising; thus, less

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Fig. 2. Coupled HSI denoising and unmixing.

spectral distortion will be caused, and better denoising result will be obtained. On the other hand, by using the better denoising result, the unmixing will be more robust to noise, and it will lead to a better interpretation result. Given an initialization, the coupled scheme solves iteratively the denoising problem with the constraint of the unmixing result. In each iteration step, it alternates between denoising and unmixing, with the denoising and unmixing results achieved simultaneously. It is worth noting that if denoising and interpretation are formulated in a unified framework, coupling denoising and interpretation is possible. The coupled scheme offers a general model to deal with different tasks of HSI restoration and interpretation, such as denoising–unmixing, deblurring–classification, and superresolution–target detection.

ˆ = arg min 1 Y − AS2F + τ S1,1 S 2 S

III. C OUPLE D ENOISING AND U NMIXING

s.t.

S>0 (2)

Since both of denoising and unmixing can be formulated and implemented in sparse representation framework, it would be possible to couple them in a closed loop by connecting them in the same sparsity framework. In the following, we will describe the CHyDU algorithm in a sparse representation framework. A. Unmixing Based on Sparse Regression Spectral mixture is unavoidable due to the limited spatial resolution of the hyperspectral imager. A linear mixture model (LMM) assumes that the observed spectrum on each pixel is the linear combination of a few pure spectral signatures (endmembers) [4], [33], i.e., ¯ +W Y = ES

with each column representing the spectral signature of one endmember. S ∈ RP ×MN is the abundance matrix; its element sp,i represents the fractional abundance of the pth endmember on the ith pixel. Unmixing is a procedure to determine the endmember abundance on each pixel. In general, the number of endmembers on each pixel is limited and far less than the number of spectral endmembers contained in a spectral library (e.g., USGS spectral library). If a spectral library is given, the spectrum on every pixel can be represented linearly and sparsely using few endmembers in the spectral library. Taking the sparsity as regularization, an unmixing method based on sparse regression is presented as follows [33]:

(1)

where Y = [y 1 , y 2 , . . . , y n ] ∈ RL×MN , and the column of Y is the spectrum on every pixel. M , N , and L are the number of rows, columns, and spectral bands of the HSI, respectively. W ∈ RL×MN is the measurement noise or model error. Matrix E ∈ RL×P contains the spectral signatures of P endmembers,

is any given spectral library, and it contains where A ∈ R the spectral signature of m(m  P ) endmembers. || · ||1,1 is L1,1 -norm, which sums the absolute value of all the elements in S; it is the sparsity regularizer for the abundance S. The Frobenius norm || · ||F forces the regression error to be low. Parameter τ balances the tradeoff between the regression error and sparsity. S > 0 means that the abundance value should be nonnegative. It is worth noting that A ∈ RL×m is different to E ∈ RL×P in (1). In the sparse-regression unmixing method, A is a given spectral library containing spectral signatures of a large number of materials; only a few of these materials are present in the scene. E is a subset of A; it only contains the spectral signatures of materials present in the scene. Both S ∈ Rm×MN in (2) and S ∈ RP ×MN in (1) denote abundance matrix, but they have different sizes. S is the abundance matrix corresponding to endmembers in A; S is the abundance matrix corresponding to the endmembers in E. Most elements in matrix S are zero or close to zero due to the sparsity prior. The given unmixing algorithm is called the Sparse Unmixing via variable Splitting Augmented Lagrangian method (SUnSAL method) [33]. In [31], based on the SUnSAL method, L×m

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the TV of the abundance map is added as a regularizer to exploit the spatial correlation, i.e., ˆ = arg min 1 Y − AS2F + τ S1,1 S 2 S + τTV TV(S) s.t. S > 0. (3)   TV(S) = i j∈ε ||si − sj ||1 is the nonisotropic TV of the abundance map, si is the abundance vector on the ith pixel, and ε denotes the set of horizontal and vertical neighbors of the ith pixel in the image. TV promotes the piecewise smooth transitions by requiring the abundance of the same endmembers on neighboring pixels to be close to each other. This algorithm is denoted as SUnSAL-TV in the following.

For HSIs, redundancy and correlation in global spectral dimension should be exploited to improve denoising performance. In this paper, we exploit the redundancy and correlation of HSIs based on a low-rank assumption. As proved in [11], under the spectral linear mixture assumption, a 3-D HSI can be reshaped into a 2-D matrix by vectorizing image in each band. Then, the rank of the clean HSI (2-D matrix) is low and far less than its size, whereas the noise HSI does not have this property due to the statistical characteristic of noise. The lowrank a priori property of the clean HSI is exploited by adding a low-rank regularizer to the sparsity-based denoising objective function, i.e.,   ˆ D, ˆ {α X, ˆ i } = arg min γX − Y 2F X,D,αi

+

B. Coupled Denoising and Unmixing

I  

 Ri X − Dαi 2F + ηαi 1 + μX∗

(5)

i=1

HSI denoising can also be formulated and implemented in a sparse representation framework [11], [20]. In the sparse representation framework, under a Gaussian white noise model, the dictionary can be trained from HSI, and every patch of HSI can be reconstructed with few atoms sparsely and linearly, i.e.,   ˆ D, ˆ {α X, ˆ i } = arg min γX − Y 2F X,D,αi

+

I    Ri X − Dαi 2F + ηαi 0

(4)

i=1

where X ∈ RL×MN is the noise-free HSI and Y ∈ RL×MN is the noisy HSI, both are 2-D matrices transformed from the original 3-D version [11]. M and N are the number of rows and columns of the image, respectively. L is the number of spectral bands. Ri is the operator that extracts the ith overlapping patch Ri X from X. I is the number of image patches. D is the trained dictionary (it can also be generated using wavelet basis), where the column of D is called the atom, and αi is the representation coefficient vector of Ri X over the dictionary. Only a few elements of αi are nonzero, L0 -norm || · ||0 computing the number of nonzero elements acts as sparsity constraint, and the Frobenius norm || · ||F forces the approximation error to be low. L0 -norm is a pseudonorm and not convex; the uniqueness of solution cannot be guaranteed. At the same time, L0 -minimization is an NP-hard combinatorial search problem; therefore, L0 -norm is replaced with L1 -norm in the following, which is the closest convex relaxation to L0 -norm. L1 -norm is the sum of the elements’ absolute value and is a convex function; thus, the uniqueness of the solution can be guaranteed. The sparse solution of the L1 -relaxed problem can approximate the original L0 -minimization well [45]. What is more, the L1 -relaxed problem can be efficiently solved by many off-the-shelf methods [46]. The first term in (4) helps to remove visible artifacts on patch boundaries, where the weight γ depends on the variance of noise.

where || · ||∗ in the last term in (5) represents nuclear norm, which sums the absolute value of all the singular values [21], [29]. It forces the rank of noise-free HSI to be low. To improve on the traditional scheme where denoising and unmixing are treated separately, the unmixing result is used as feedback in a closed-loop way. If endmembers are correctly selected and their corresponding unmixed abundances are provided with high accuracy, we can expect that the spectrum on any pixel obtained from the unmixing result is close to that of the clean HSI. Based on this a priori, we introduce a Frobenius norm term ||X − AS||2F , where A and S represent the spectral library and the abundance respectively, as the disparity between pixel spectra and expected clean HSI. It acts as a spectral regularizer to enhance spectral fidelity. Equation (5) becomes   ˆ D, ˆ {α X, ˆ i } = arg min γX − Y 2F X,D,αi

+

I  

Ri X − Dαi 2F + ηαi 1



i=1

+ μX∗ + τ 1 X − AS2F .

(6)

Parameter τ1 is the weight of the spectral regularizer. It is assumed that the abundance S is given in the above objective function. In reality, the abundance S is not known. Here, the abundance S is obtained via sparse coding by employing its sparsity prior over a chosen spectral library A; therefore, the L1,1 -norm of S is added as follows:   ˆ D, ˆ {α ˆ = arg min γX − Y 2F X, ˆ i }, S X,D,αi ,S

+

I    Ri X − Dαi 2F + ηαi 1 i=1

+ μX∗ + τ1 X − AS2F + τ2 S1,1

s.t.

S ≥ 0.

(7)

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The nonnegative constraint is enforced on S. Objective function (7) is proper and coercive, and it is convex with respect to each variable; therefore, the existence and uniqueness of the solution of (7) can be guaranteed [35]. In order to solve (7) conveniently, we introduce an auxiliary variable U to split X. Variable X in the fourth term is replaced by U with constraint X = U ; then, (7) is changed to the following:   ˆ U ˆ = arg min γX − Y 2F ˆ D, ˆ {α X, ˆ i }, S, X,D,αi ,S,U

+

I  

Ri X −Dαi 2F +ηαi 1



i=1

s.t.

S ≥ 0, X = U . (8)

Based on the augmented Lagrangian method, we solve the following unconstrained counterpart for a proper parameter ρ:   ˆ D, ˆ {α ˆ U ˆ = arg min γX − Y 2F X, ˆ i }, S, X,D,αi ,S,U

+

I    Ri X −Dαi 2F +ηαi 1 i=1

+ μU ∗ + τ 1 X − AS2F  + τ 2 S1,1 + tr λT (X − U ) + ρX − U 2F

s.t.

S≥0

optimize (9) with respect to one variable by fixing others in each step; thus, the optimization problem reduces to the following subproblems. 1) Solving D and αi by fixing U , X, and S, i.e., I      ˆ {α D, ˆ i } = arg min Ri X −Dαi 2F +ηαi 1 . D,αi

(9)

where λ ∈ RL×MN is the Lagrange multiplier matrix, and tr(·) is trace of matrix. If parameter ρ is set properly, problems (8) and (9) would be equivalent [27]. The solving method of (9) will be presented in the following. It should be noted that the HSI X is modeled in a sparse representation framework, and the second term of (7) is the data fidelity term. The denoising problem is an underdetermined inverse problem; therefore, regularization is necessary. The unmixing result AS is introduced to suppress the spectral distortion of the denoising result X but not to model the denoising result X. The fifth term of (7) only acts as regularization to reduce the ill-posedness of the solution of X. Both of the abundance and the denoising HSI are unavailable, and (7) is defined with respect to variable X, D, αi , and S. All of these variables can be solved simultaneously. Although it ˆ =D ˆ ◦α seems that both the sparse coding result (i.e., X ˆ i) ˆ ˆ and the unmixing result (i.e., X = AS) can be treated as a denoising result, only the result achieved by minimizing (7) with respect to X is optimal because it combines advantages of noise reduction from sparse coding and spectral fidelity from unmixing. C. Solver There are more than one variable to be determined in (9); it can be solved via alternative optimization [27], [37]. We

i=1

(10) It can be reduced to the following: ˆ = arg min D D

+ μU ∗ + τ 1 X − AS2F + τ 2 S1,1 ,

5

I 

Ri X − Dαi 2F

(10a)

i=1

α ˆ i = arg minRi X −Dαi 2F +ηαi 1 , αi

for i = 1, 2, . . . , I.

(10b)

This step is dictionary learning and sparse coding, corresponding to (10a) and (10b), respectively. The K-means singular value decomposition (K-SVD) method can be used to update the dictionary D [28], [32], and the sparse coefficient αi can be solved using existing L1 − L2 minimization methods. Although the recovered image X is unavailable and should be initialized with noisy measurement Y in the first iteration, we point out that dictionary learned from Y can reconstruct X well because noise is implicitly rejected to some extent in dictionary learning [20]. 2) Solving S by fixing U , X, D, and αi ˆ = arg min 1 X − AS2F + τ2 S1,1 s.t. S ≥ 0. S 2 2τ1 S (11) This step is sparse-regression-based unmixing. We solve it using the alternative direction method of multiplier [27]. 3) Solving U by fixing X, D, αi , and S  ˆ = arg min μ U ∗ + 1 tr λT (X − U ) U 2ρ 2ρ U 1 + X − U 2F . (12) 2 After a straightforward complete-the-squares processing, the second and third terms can be incorporated into a single quadratic term (plus a constant independent of U ), leading to the following alternative form:

 2

1 μ 1

ˆ U = arg min U ∗ + U − X + λ

. 2ρ 2 2ρ U F (13) ˆ = The solution of the given problem is U W 1 · soft(W 2 , μ/2ρ) · W 3 , where svd(X + λ/2ρ) = W 1 · W 2 · W 3 is the SVD of matrix X + λ/2ρ. soft(W 2 , μ/2ρ) means applying elementwise softthreshold shrinkage to singular matrix W 2 ; the shrinkage threshold is μ/2ρ.

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4) Solving X by fixing U , D, αi , and S

ˆ = arg min γX − Y 2F + X X

I 

Algorithm 1 Coupled HSI Denoising and Unmixing (CHyDU-De & CHyDU-Un)

Ri X − Dαi 2F

i=1

 +τ1 X −AS2F +tr λT (X −U ) +ρX −U 2F . (14) Following the same complete-the-squares strategy as in solving (12) and (14) can be rewritten as:

ˆ = arg min γX − Y 2F + X X

I 

Input: noisy HSI Y , spectral library A Output: denoised HSI X, abundance S Initialization: X 0 = Y , λ0 = 0, k = 1, maximum number of iterations K_ max while k < K_ max do Step 1. Dictionary learning and sparse coding I   k    2 ˆ , α D Ri X k−1 −Dαi F ˆ ki = arg min

Ri X − Dαi 2F

i=1

D,αi

 2



1

. (15) X − U + λ + τ1 X − AS2F + ρ



2ρ F

ˆ = X

I 

 + ηαi 1 .

Step 2. Solving abundance S

It has a closed-form solution, i.e.,

i=1

ˆk = arg min 1 X k−1 − AS2 S F 2 S τ2 + S1,1 , s.t. S > 0. 2τ1

−1 Ri T Ri + γI + τ1 I + ρI

i=1

  1 · Ri Dαi + γY + τ 1 AS + ρ U + λ . 2ρ i=1  I 

Step 3. Solving auxiliary variable U ˆ k = arg min μ U ∗ U 2ρ U

2  1

1 k−1

k−1

. + U − X + λ

2 2ρ F

(16) The Lagrange multipliers matrix is initialized with zero; in the kth iteration, it is updated according to λk = λk−1 + 2ρ(X k − U k ) [27]. According to [42, Th. 8], which is a study on convergence, if the following two conditions are satisfied, then the solutions of subproblems (10)–(12) and (14) are certain to converge to that of (7). First, the matrix from which variable X in the fourth term in (7) is splitted is of a full column rank. Second, the subproblems (10)–(12) and (14) are solved exactly. The first condition is satisfied as, in this paper, it is an identity matrix and hence is of full column rank. As for the second condition, it is worth noting that even if parts of the subproblems are not in closed-form solution and cannot be solved exactly in strict sense, as long as the errors at each iteration decrease progressively and are absolute summable, the convergence is still guaranteed [27]. In this paper, subproblems (12) and (14) have closed-form solutions. Subproblems (10) and (11) have no closed-form solutions, but their errors are decreasing and absolute summable [28], [32], [37]; therefore, the second condition is still met. Since the above two conditions are met, the convergence of the solver can be guaranteed. We repeat the optimization process of the given subproblems until the following stopping criterion is met. The algorithm stops when the norm of difference between the solutions of two consecutive iterations is smaller than a given positive constant or it reaches a predefined iteration number. The proposed CHyDU algorithm is summarized as Algorithm 1.

Step 4. Solving denoised HSI X

I −1  k T ˆ = Ri Ri + γI + τ1 I + ρI X i=1

·

 I 

Ri D k αki + γY + τ1 AS k

i=1

  1 +ρ U k + λk−1 . 2ρ Step 5. λk = λk−1 + 2ρ(X k − U k ), k = k + 1 end D. Computational Complexity For the procedures mentioned in Algorithm 1, the computation of step 1 involves only sparse coding. If dictionary D ∈ Rn×m is given, for a signal of length n, the computational cost of sparse coding is O(Kmn), where K is the sparsity, i.e., a number of nonzero coefficients over dictionary, is assumed a constant, and is far smaller than n. If the size of HSI is M × N × L, sparse coding has a computational cost of O(M N L). In step 2, abundance S k can be solved in O(M N L2 )operations [33]. Step 3 involves matrix SVD decomposition and threshold

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shrinkage. SVD decomposition of X k−1 can be computed in O(M 2 N 2 L) operations. Because W 2 is a diagonal matrix, the computational complexity of the elementwise soft shrinkage is only O(L). Variable U k can be computed at the cost of O(M 2 N 2 L). Although expression of step 4 is cumbersome, it only shows averaging of all the denoised patches with a small portion of Y and U k . The cost of computing X k is O(M N L). Thus, the overall computational complexity of our method per iteration is O(M 2 N 2 L + M N L2 ). IV. E XPERIMENT R ESULT AND A NALYSIS To test our method, we use two simulated HSIs and one real HSI. Comparisons are made between our method and state-of-the-art methods. The compared denoising methods are the BM4D method [30], [34], the PCA+Shrink method [8], and our former proposed Spa+Lr method [11]. BM4D and PCA+Shrink methods are state-of-the-art methods in both accuracy and speed. The compared unmixing methods are FCLS method [39], SUnSAL method [33], and SUnSAL-TV method [31]. The SUnSAL and SUnSAL-TV methods are in a sparse representation framework and require a given spectral library as same as our method; this leads to a fair comparison. The denoising result of our method is denoted CHyDU-De; for the unmixing result, it is denoted CHyDU-Un. Quantitative assessing indexes to evaluate the denoising performance include peak-signal-to-noise ratio (PSNR, dB), structural similarity index measurement (SSIM), and feature similarity index measurement (FSIM) [47], [48]. For each of the indexes, we present the mean values over all the bands. The unmixing performance is assessed using signal reconstruction error (SRE, dB), which is defined as [31]   ˆ 2F (17) SRE = 10 log S2F /S − S ˆ is the estimated where S is the reference abundance, and S abundance. A. Result on Simulated HSI Two simulated HSIs are generated here. The abundance maps of simulated HSI 1 contain several square blocks. The abundances of simulated HSI 2 follow Dirichlet distribution uniformly over the probability simplex (see Figs. 3–8 for details on simulated HSI 1 and Figs. 9–14 for details on simulated HSI 2). Based on LMM, the abundance matrix is multiplied by spectral signatures chosen randomly from the USGS spectral library to generate the simulated HSIs. Both of the simulated HSIs are provided by Dr. M. D. Iordache of Extremadura University. Size of simulated HSI 1 is 75 × 75 × 224, and it contains five endmembers. Size of simulated HSI 2 is 100 × 100 × 224, and it contains nine endmembers. The reference abundance maps are shown in Figs. 3 and 9. Noise at different levels is added to the simulated HSI. The standard variance of noise varies from 0.025 to 0.250 to model both low and heavy noise cases. It is worth noting that, for real images, the initial noise levels are different from band to band, but a standard practice is to preprocess the HSI to make noise

Fig. 3. Abundance maps of simulated HSI 1.

level the same across all the bands using whitening technique [11]. In our experiments, the same levels of noise are added to all different bands. Table I lists the quantitative assessments of denoising on simulated HSI 1, and Table II lists that of unmixing. The assessment indexes show that the proposed CHyDU algorithm can provide simultaneously better denoising results and unmixing results than the results from other methods. Most of the indexes of denoising and unmixing are higher than the state-of-theart methods. Similar observations can be made for the results generated from simulated HSI 2 (see Tables IV and V). BM4D is one of the state-of-the-art denoising methods. The mean PSNR indexes in Tables I and IV indicate that CHyDUDe denoising result is competitive to BM4D in most cases. BM4D performs better on simulated HSI 1 than on simulated HSI 2. It outperforms CHyDU-De when σ = 0.200 and σ = 0.250 for simulated HSI 1. BM4D reduces noise via utilizing the similarity in spatial and spectral dimensions, and it works well when structural similarity is high. As simulated HSI 1 contains regular blocks, it has higher spatial similarity than simulated HSI 2; therefore, BM4D performs better on simulated HSI 1 than on simulated HSI 2. CHyDU-De is actually an extended version of Spa+Lr in [11] with the introduction of the spectral regularizer ||X − AS||22 in a closed-loop scheme, which constraints the spectral signature of the denoised data to be close to the result of unmixing. This leads to better spectral fidelity; therefore, CHyDU-De achieve better denoising result compared with Spa+Lr. In order to assess the visual quality of denoising, the mean SSIM and mean FSIM indexes are used (see Tables I and IV). These two indexes are designed based on human vision characteristics. High index values mean better resemblance to the clean image in visual sense. In the σ = 0.200 and σ = 0.250 case, with lower mean FSIM values, the denoised images of BM4D are much blurred and have poorer visual quality; even the mean PSNR values are higher. The mean SSIM and mean FSIM of CHyDU-De are often the highest, indicating superior denoised results of this method. Parts of the denoised images are shown in Figs. 4, 5, 10, and 11. The BM4D method first searches image blocks with similar spatial and spectral features. Blocks with similar spatial–spectral features are denoised collaboratively, leading

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Fig. 4. Denoised images of simulated HSI 1 in band 1. (a) Original noise-free image. (b) Noisy version of σ = 0.100. (c) Result of BM4D. (d) Result of PCA+Shrink. (e) Result of Spa+Lr. (f) Result of the proposed CHyDU-De method.

Fig. 5. Denoised images of simulated HSI 1 in band 5. (a) Original noise-free image. (b) Noisy version of σ = 0.250. (c) Result of BM4D. (d) Result of PCA+Shrink. (e) Result of Spa+Lr. (f) Result of the proposed CHyDU-De method.

Fig. 6. PSNR values in each spectral band of denoised simulated HSI 1 under different noise levels: σ = 0.025 in the left, and σ = 0.100 in the right.

to reduced differences in these image blocks. The result is that the denoised image would be blurred, as shown in Figs. 4, 5, 10, and 11. Comparatively, the residual noise of CHyDU-De is less than that of the other denoising methods, as shown by the higher PSNR values in Figs. 6 and 12; this demonstrates the effectiveness of our method. Due to the limitation of space, only parts of the unmixing results can be shown (see Figs. 8 and 14). When the sparsity of abundance over the spectral library is exploited, as the case of

SUnSAL, better unmixing result would be obtained. With the increase in noise level, the performance of FCLS and SUnSAL decreases. SUnSAL-TV exploits spatial correlation to generate high spatial consistency in abundance maps even in high noise cases. However, the TV term in SUnSAL-TV also enforces neighboring pixels to have similar endmembers and abundance. As a result, blurring occurs and smaller details are missing, as observed in Figs. 8 and 14. In the proposed CHyDU-Un algorithm, unmixing and denoising are implemented in a coupled

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Fig. 7. Spectral distortion on pixel (10, 20) of denoised simulated HSI 1 under different noise levels: σ = 0.100 in the left, and σ = 0.200 in the right.

Fig. 9. Abundance maps of simulated HSI 2.

Fig. 8. Unmixing result of simulated HSI 1 when σ = 0.025, abundance maps of endmembers 1, 2, and 3 are shown. From top to bottom: ground truth of abundance map, and unmixing results of the FCLS, SUnSAL, SUnSAL-TV, and proposed CHyDU-Un methods, respectively.

way where noise is suppressed in sparse representation in each iteration. As a result, the unmixing result is enhanced further. The CHyDU-Un results show that it is robust to noise. CHyDU-Un does not use any strict spatial constraint such as TV; therefore, more details and edges are preserved. Compared with the oversmoothed SUnSAL-TV result, the abundance map of CHyDU-Un is sharper, as shown in Figs. 8 and 14. The spectral reflectance is critical for the application of HSIs. Spectral distortion should be avoided in denoising. Here, the spectral distortion is evaluated using spectral angle (SA, in degrees) [41]. After computing the SA values of all pixels, we compute the mean value, denoted MSA (in degrees). The MSA values of denoised results under different noise levels

on simulated HSI 1 and simulated HSI 2 are reported in Tables III and VI, respectively. The MSA values of CHyDU-De are lower than other denoised results in most cases. Spectral correlation in global dimension is exploited in Spa+Lr via a low-rank constraint, which helps to restrain spectral distortion. In the CHyDU-De method, not only is the low-rank constraint exploited but the endmember abundance is also used as spectral regularizer, which leads to better spectral fidelity. Other than the MSA, the spectrum difference curve between noise-free and denoised pixels is also generated to assess spectral distortion. The spectrum difference curves on pixel (10, 20) of simulated HSI 1 under σ = 0.100 and σ = 0.200 are given in Fig. 7, and the spectrum difference curves on pixel (10, 10) of simulated HSI 2 under σ = 0.075 and σ = 0.100 are given in Fig. 13. From these curves, we can observe that the spectrum difference of our method is closer to zero. On simulated HSI 1, when σ = 0.100, the mean values of spectrum difference on pixel (10, 20) by BM4D, PCA+Shrink, Spa+Lr, and CHyDUDe are 0.0162, 0.0123, 0.0127, and 0.0036, respectively. When σ = 0.200, they are 0.0181, 0.0245, 0.0142, and 0.0100, respectively. From the experiments on simulated HSI 2, when σ = 0.075, the mean values of spectrum difference on pixel (10, 10) by BM4D, PCA+Shrink, Spa+Lr, and CHyDU-De

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Fig. 10. Denoised images of simulated HSI 2 in band 5. (a) Original noise-free image. (b) Noisy version of σ = 0.075. (c) Result of BM4D. (d) Result of PCA+Shrink. (e) Result of Spa+Lr. (f) Result of the proposed CHyDU-De method.

Fig. 11. Denoised images of simulated HSI 2 in band 20. (a) Original noise-free image. (b) Noisy version of σ = 0.250. (c) Result of BM4D. (d) Result of PCA+Shrink. (e) Result of Spa+Lr. (f) Result of the proposed CHyDU-De method.

Fig. 12. PSNR values in each spectral band of denoised simulated HSI 2 under different noise levels: σ = 0.025 in the left, and σ = 0.075 in the right.

are 0.0172, 0.0185, 0.0165, and 0.0122, respectively. When σ = 0.100, the mean values of the spectrum difference are 0.0216, 0.0243, 0.0196, and 0.0140, respectively. These results indicate that our method performs better. From the experiments on simulated HSI 2, we can find that the assessing indexes of CHyDU in Tables IV and V are also higher than the other denoising methods and unmixing methods, the visual superiority of denoised images and abundance

maps is also obvious, and the spectral distortion in Table VI is also less than the others, which validates the effectiveness of our proposed CHyDU method further. B. Convergence Experiment The iteration process is presented in Fig. 15, which shows the mean PSNR value in each iteration. The denoising performance

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Fig. 13. Spectral distortion on pixel (10, 10) of denoised simulated HSI 2 under different noise levels: σ = 0.075 in the left, and σ = 0.100 in the right.

Fig. 14. Unmixing results of simulated HSI 2 when σ = 0.075. Abundance maps of endmembers 1, 3 and 5 are shown. From top to bottom: ground truth of abundance map and unmixing results of FCLS, SUnSAL, SUnSAL-TV, and the proposed CHyDU-Un method, respectively.

represented by mean PSNR peaks after three iterations; therefore, three iterations are adequate to achieve good results. After three iterations, mean PSNR plateaus. If the iteration number is too large, denoising result may deteriorate because of overfitting in the sparse coding and unmixing stage, which will result in oversmoothing.

As discussed in the end of Section III-C, the convergence of the proposed method is guaranteed. In order to further demonstrate the convergence is indeed secured, we also give the evolution of the objective function value over the iterations in Fig. 16. The objective function value decreases with each iteration, but after the big descend in the second iteration, only

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TABLE I A SSESSING I NDEXES OF THE D ENOISED R ESULTS U NDER N OISE OF D IFFERENT L EVELS ON SIMULATED HSI 1

TABLE IV A SSESSING I NDEXES OF THE D ENOISED R ESULTS U NDER N OISE OF D IFFERENT L EVELS ON SIMULATED HSI 2

TABLE II SRE VALUE OF U NMIXING R ESULTS U NDER N OISE OF D IFFERENT L EVELS ON SIMULATED HSI 1

TABLE V SRE VALUE OF U NMIXING R ESULTS U NDER N OISE OF D IFFERENT L EVELS ON SIMULATED HSI 2

TABLE III MSA ( IN DEGREES ) OF THE D ENOISED R ESULTS U NDER N OISE OF D IFFERENT L EVELS ON SIMULATED HSI 1

very small margins are recorded before halting at a constant. This illustrates the convergence of the proposed method. C. Parameter Setting and Sensitivity Analysis 1) Parameter Setting for CHyDU: For our CHyDU method, there are six regularization parameters to tune. The first is the weight of the denoising fidelity term γ; it determines the weight of Y when we solve X in (16). We set it γ = 1/8.5σ according to [20]. The second one is η, which balances the representation error and sparsity. It depends on the noise level

TABLE VI MSA ( IN D EGREES ) OF THE D ENOISED R ESULTS U NDER N OISE OF D IFFERENT L EVELS ON SIMULATED HSI 2

and is handled implicitly in the process of sparse coding in (10) with the augmented Lagrangian method. Satisfying results can be acquired by setting the representation error to 1.15σ [32]. The third is regularization parameter μ, which weights the lowrank constraint and is dependent on the noise variance and data size [29], [40]. The ratio μ/2ρ is the shrinkage threshold of the singular value of X + λ/2ρ when solving the auxiliary variable U in (13). If the size of X is large and the noise level is high, the singular value of X will be large, and it should be shrunk with a relative high √ In [29], the ratio μ/2ρ is √ threshold. suggested to be set to c( M N + L)σ, reminded that M and

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Fig. 15. Relations between the iteration numbers and denoising performance. From left to right: σ = 0.05, σ = 0.10, σ = 0.15, σ = 0.20 and σ = 0.25. Top: results on simulated HSI 1; bottom: results on simulated HSI 2.

Fig. 16. Evolution of the objective function value over the iterations: σ = 0.050 on simulated HSI 1 in the left, and σ = 0.150 on simulated HSI 2 in the right.

N are the rows and columns of the images, L is the number of √ bands, and c is a given constant. Considering that M N  √ L in the general √ case, the ratio between μ and ρ can be reduced to μ/ρ = σ M N /6.5. As the noise level is expected to decrease during the optimization process, parameter μ is updated iteratively. In the experiment, we divide this parameter by 10 in each iteration loop. The fourth parameter is τ1 , which is the weight of the spectral regularizer. It determines the weight of AS when we solve X in (16). Our experiments show that the performance of CHyDU is insensitive to this parameter, and we set it to 20. The fifth parameter τ2 controls the sparsity of abundance. The ratio τ2 /2τ1 in (11) is dependent on the noise variance. In the following experiment, the noise variance is [0.0250, 0.050, 0.075, 0.100, 0.150, 0.200, 0.250] to model the light and heavy noise cases; then, for simulated HSI 1, τ2 /2τ1 is set to [0.05, 0.2, 0.5, 0.75, 1, 1.5, 2], and for simulated HSI 2, they are [0.01, 0.05, 0.1, 0.15, 0.25, 1.2, 1.5]. The sixth parameter ρ indicates that the auxiliary variable U should be close to the estimated X. It determines the weight of auxiliary variable U in solving X in (16). The augmented Lagrangian method does not impose any constraint on parameter ρ. It is generally difficult to determine theoretically how large the ρ value must be. It can be found that both of high accuracy and fast convergence can be obtained when parameter ρ is around 100; therefore, we keep ρ = 100. 2) Parameter Setting for Compared Methods: The parameter setting of the compared algorithms first follow suggestions from the authors, and then they are empirically tuned to achieve the best result. We adopted all default values for parameter setting of the BM4D algorithm. The only parameter for the PCA+Shrink method is the number of principal components,

where the setting of 4 gives the best results. The sparsity framework of Spa+Lr is the same as the proposed CHyDU method; therefore, all the parameter setting are the same as discussed in Section IV-C1. There is no parameter to tune for FCLS. For the SUnSAL method, we have to set the weight of sparsity according to the noise variance varying at [0.025, 0.050, 0.075, 0.100, 0.150, 0.200, 0.250]. They are [0.05, 0.2, 0.5, 0.75, 1, 1.5, 2] for simulated HSI 1 and [0.01, 0.05, 0.1, 0.15, 0.25, 1.2, 1.5] for simulated HSI 2. There are two parameters for the SUnSAL-TV method: weight of sparsity and weight of the TV regularizer. Using the same seven noise variance levels as listed above, the parameter pair is set as [(0.01, 0.02), (0.01, 0.05), (0.01, 0.075), (0.02, 0.1), (0.05, 0.2), (0.1, 0.2), (0.2, 0.2)] for simulated HSI 1 and [(0.01, 0.002), (0.005, 0.01), (0.05, 0.01), (0.02, 0.02), (0.02, 0.05), (0.02, 0.1), (0.02, 0.2)] for simulated HSI 2. 3) Sensitivity Analysis: There are six regularization parameters involved in the solving of objective function (9); we have discussed the selection process of five of them in Section IV-C1, and the remaining spectral regularizer weight τ1 is the only parameter to be tuned. To present the basis for the tuning of τ1 , we give the sensitivity analysis of the proposed CHyDU method over τ1 . It is found that the denoising and unmixing performance do not change significantly with τ1 at the range of 0.1 to 100, which proves the robustness of the CHyDU method over τ1 . In Figs. 17 and 18, we present the mean PSNR and SRE indexes corresponding to varying τ1 . It is obvious that changes of these indexes are insignificant. Since the proposed method is not sensitive to τ1 , we concluded τ1 can be set as a fixed value. It is worth noting that, in the varying of τ1 , the value of τ2 was also varied to guarantee that τ2 /(2τ1 ) was kept a constant, which is selected according to the principle given in Section IV-C1.

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Fig. 17. Sensitivity analysis between the denoising performance and parameter τ1 , σ = 0.050 on simulated HSI 1 in the left, and σ = 0.200 on simulated HSI 2 in the right.

Fig. 18. Sensitivity analysis between the unmixing performance and parameter τ1 , σ = 0.100 on simulated HSI 1 in the left, and σ = 0.150 on simulated HSI 2 in the right.

Fig. 20. Denoised images of Cuprite data in band 136. (a) Noisy measurement.(b) Result of BM4D. (c) Result of PCA+Shrink. (d) Result of Spa+Lr. (e) Result of the proposed CHyDU-De method.

Fig. 19. Synthetic color map of Cuprite data, the marked region is the experiment data.

D. Result on Real HSI The given experiments are performed on simulated HSIs. However, in real case the noise level varies with the change of spectral band. Here, we consider denoising and unmixing in real noisy hyperspectral data to verify the validity of the proposed CHyDU method. The HSI that we use is Cuprite,2 which is captured by the AVIRIS sensor over Cuprite mine district in Nevada. These data have 224 spectral bands between 400 and 2 Available

online: http://aviris.jpl.nasa.gov/html/aviris.freedata.html.

2500 nm. Images in band 1–2, 104–113, 148–167, and 221–224 are abandoned due to water absorption or low SNR. Part of the scene (in the red marked region of Fig. 19) is extracted as experiment data; thus, the size of the data is 150 × 130 × 188. Fig. 19 is the synthetic color map3 of the Cuprite data. Because there is no ground truth of denoising and unmixing for real HSIs, we cannot give any assessing index here as in the simulated HSI experiment. Denoised images in band 136 are shown in Fig. 20. There is still some noise that remained in the result of Spa+Lr. The residual noise in the result of CHyDU-De is less than result of Spa+Lr; at the same time, the edges and details in the result of CHyDU-De are preserved well. The abundance map of three endmembers is presented in Fig. 21. 3 Available online: image.a.tmp.gif.

http://speclab.cr.usgs.gov/PAPERS/cuprite.clark.93/

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result demonstrates the superiority of the proposed method as compared with the other methods. However, since both dictionary learning and sparse coding are involved in the algorithm, it is computationally costly. To accelerate the speed of this algorithm is one of our future research directions. ACKNOWLEDGMENT The authors would like thank the editors and the reviewers for their insightful comments and constructive suggestions that significantly improve this paper. R EFERENCES

Fig. 21. Unmixing result of Cuprite data, abundance maps of endmembers 141, 202, and 236 are shown. From top to bottom: spectral reflectance of corresponding endmembers, and abundance maps of the FCLS, SUnSAL, SUnSAL-TV, and proposed CHyDU-Un methods, respectively.

It is clear that the proposed CHyDU-Un method can suppress outliers and isolated points that may be caused by noise, and the abundance maps of CHyDU-Un have better spatial consistence. All of the experiments are implemented in MATLAB 7.8.0 with an Intel personal computer Core 3.10 GHz, RAM of 8 GB. The BM4D method takes about 90 s, the PCA+Shrink method takes about 25 s, the Spa+Lr method takes about 3 h, the FCLS method takes about 50 s, the SUnSAL method takes 15 s, the SUnSAL-TV method takes about 130 s, and the proposed CHyDU method takes about 3 h to obtain the denoising and unmixing results. Most of the computation time of the CHyDU method is for the sparse coding of each image patch. Fortunately, sparse coding on every patch is independent of each other; therefore, it can be accelerated in a parallel way, and then the running time will decrease drastically. V. C ONCLUSION In this paper, we point out that hyperspectral denoising and unmixing can regulate and promote each other. Through formulating them in a unified sparse representation framework, we designed a coupled denoising and unmixing algorithm for HSIs where denoising and unmixing results are obtained simultaneously. Based on the idea that disparity between spectrum obtained from a good unmixing and clean HSI should be small, abundance and endmembers are being utilized as spectral regularizers to restrain spectral distortion in sparse denoising. The better denoising result further improves sparse unmixing, and the unmixing result is robust to noise. The experiment

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Jingxiang Yang (S’14) received the B.S. degree in automation from Anhui Polytechnic University, Wuhu, China, in 2011 and the M.S. degree in control theory and control engineering from Northwestern Polytechnic University, Xi’an, China, in 2014. He is currently working toward the Ph.D. degree with the School of Automation, Northwestern Polytechnic University. His research interests include hyperspectral image processing, sparse representation, and deep learning.

Yong-Qiang Zhao (M’05) received the B.S., M.S., and Ph.D. degrees in control science and engineering from the Northwestern Polytechnic University, Xi’an, China, in 1998, 2001, and 2004, respectively. From 2007 to 2009, he worked as a Postdoctoral Researcher at McMaster University, Hamilton, ON, Canada, and Temple University, Philadelphia, PA, USA, respectively. He is currently a Professor with Northwestern Polytechnical University. His research interests include polarization vision, hyperspectral imaging, compressive sensing, and pattern recognition.

Jonathan Cheung-Wai Chan received the Ph.D. degree from The University of Hong Kong, Hong Kong, in 1999. From 1998 to 2001, he was a Research Scientist with the Department of Geography, University of Maryland, MD, USA. From 2001 to 2005, he was with the Interuniversity Micro-Electronics Centre, Leuven, Belgium. From 2005 to 2011, he had been with the Department of Geography, Vrije Universiteit Brussel (VUB), Brussels, Belgium. From 2013 to 2014, he was a Marie Curie Fellow with Fondazione Edmund Mach, Italy. He is currently a Senior Researcher and a Guest Professor with the Department of Electronics and Informatics, VUB. His research interests include machine learning algorithms for land-cover classification, detailed mapping using hyperspectral data, and spatial enhancement of satellite hyperspectral images and applications.

Seong G. Kong (SM’03) received the B.S. and M.S. degrees from Seoul National University, Seoul, Korea, in 1982 and 1987, respectively, and the Ph.D. degree from the University of Southern California, Los Angeles, CA, USA, in 1991, all in electrical engineering. He was an Associate Professor with the Department of Electrical and Computer Engineering, The University of Tennessee, Knoxville, TN, USA, and with Temple University, Philadelphia, PA, USA. He also served as the Chair of the Department of Electrical Engineering, Soongsil University, Seoul, Korea, and as the Graduate Program Director with Temple University. He is currently a Professor of Computer Engineering with Sejong University, Seoul. His research interests include image processing, hyperspectral imaging, and intelligent systems.